Real Mathematical Analysis
– by Charles C. Pugh
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The three introductory Analysis books I’ve read cover-to-cover are Lang’s, Pugh’s, and Rudin’s.
What makes Pugh’s book stand out is simply that he focuses on building up repeatedly useful machinery and concepts-a broad set of theorems that are clearly motivated and widely applicable to a lot of problems. Pugh’s book is also chock-full of examples, which make understanding the material much faster. And finally, Pugh’s book has a very large number of exercises of varying difficulty-Pugh has more than 500 exercises total.
In contrast, Rudin’s book tends to focus on “magic.” Rudin uses the shortest possible proofs for a given theorem. The problem is that the shortest proofs aren’t necessarily the most instructive-while Baby Rudin is a beautiful work of Math qua Math, it’s not a particularly good book to learn from.
Finally, Lang’s book is frankly subpar. Lang leaves out critical details of some proofs (dismissing one 6 page proof as trivial!), is poorly motivated by examples, and has a number of mistakes.
If you want to really understand Mathematical Analysis and get to the point where you can use the concepts to create proofs and solve problems, Pugh is the best book on the topic. If you want a concise summary of undergraduate analysis to review, pick Rudin’s book.” – Posted by SatvikBeri at http://lesswrong.com/lw/3gu/the_best_textbooks_on_every_subject/ – by Books2Learn – 2018-02-27